where \(\gamma (t,\cdot ):\mathbb {T}\rightarrow \partial D_{t}\)  is a \(C^1\)  parametrization of the boundary of the patch and \(\mathbf {n}(t,\cdot )\)  is an outward normal vector to the boundary. We may refer to [1]}, [2]} for a detailed derivation of this equation for active scalar models. We are particularly interested in the existence of ordered structures moving without shape deformation, called V-states. More precisely, we shall focus on the existence of uniformly rotating vortex patches about their center of mass, that can be fixed at the origin, and with a constant angular velocity \(\Omega \in \mathbb {R}\) , namely
\(q(t,\cdot )=\mathbf {1}_{D_t}\quad \mbox{ with }\quad D_{t}=e^{{\rm i} t\Omega }D_{0}.\) 
